Homegrown Boro forward Josh Coburn opened the scoring following a mix-up between the West Brom back-three, as the 18-year-old smashed the ball home from close range.
Middlesbrough frustrated the Baggies as they tried to slow down West Brom's counter-attacks.
Grady Diangana broke a year long goalscoring drought as he picked up his first goal since August 2020, smashing the ball in off the post following a scramble in the Middlesbrough box.
Story of the match
Middlesbrough began the game in a robust fashion, matching the Baggies’ physically across the park.
On-loan Brighton & Hove Albion defender Matt Clarke went down with a head injury after three minutes of the game, and 16 minutes later, had to be replaced by Kyle Bartley who returned from suspension.
Albion produced some well-worked moves during the first half, but both Molumby and Callum Robinson were halted on the counter by deemed to be yellow-carded fouls.
Josh Coburn then bagged his third goal of the season, as Cedric Kipre and Conor Townsend slid in to create space for the 18-year-old who smashed home past goalkeeper Sam Johnstone.
West Brom almost gave an instant response as Snodgrass' corner evaded everyone in the six-yard box and fell to Furlong who was brought down in the area, but no foul was given by referee Matthew Donohue.
Albion boss Ismael opted for a half-time change as Robinson was hooked to be replaced by Jordan Hugill who came on to face his former club having spent a season on loan in 2018/19.
Bartley had an opportunity to score as a Furlong long throw was launched in, it was nodded on by the former Swansea City, Leeds United and Rangers man, only to be palmed away by Boro 'keeper Luke Daniels.
It was then Diangana who smashed home his first goal since August 2020 to equalise for the Baggies, the ball falling to him following a free-kick from Snodgrass.
Following a penalty shout from the Albion goalscorer, Warnock's side did their best to slow down the passages of play, captain Jonny Howson taking his time over several Boro throw-ins.
The Baggies tried to pick the lock of the Boro defence but couldn't find the all-important final ball, often looking for sideways and backwards options as Middlesbrough's defence continued to frustrate.
Man of the match - Jayson Molumby
The Irish international produced a battling performance at the heart of the Baggies midfield.
On-loan from Brighton and Hove Albion, Molumby began several of Albion's best moves going forward, completing several surging runs through the heart of the Boro midfield.
His energy and dynamism gave Ismael's side a different angle to use in midfield due to his forward-thinking efforts.